A023990 Sum of exponents of primes in multinomial coefficient M(4n; 2n,n,n) - sum of exponents of primes in M(3n; n,n,n).

0, 1, 1, 1, 2, 2, 1, 3, 3, 4, 4, 1, 2, 4, 3, 3, 5, 4, 4, 3, 4, 7, 4, 4, 5, 7, 5, 4, 5, 5, 6, 6, 7, 7, 9, 7, 9, 11, 8, 9, 9, 9, 8, 6, 7, 7, 6, 6, 8, 9, 8, 8, 9, 10, 7, 8, 9, 10, 11, 7, 8, 11, 11, 10, 12, 13, 13, 12, 11, 14, 12, 11, 13, 14, 12, 12, 14, 14, 13, 14, 14, 15, 15, 13, 14, 14, 11, 10, 13, 12, 13, 13, 12, 15

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001222, A000897, A006480, A022559.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023991, A023992, A023993.

Programs

Mathematica

a[n_] := PrimeOmega[Multinomial[2*n, n, n]] - PrimeOmega[Multinomial[n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

PARI

a(n) = bigomega((4*n)!/((2*n)!*n!^2)) - bigomega((3*n)!/(n!^3)); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)

a(n) = A023980(n) - A023978(n) = A001222(A000897(n)) - A001222(A006480(n)).

a(n) = A022559(4*n) + 2*A022559(n) - A022559(2*n) - A022559(3*n). (End)

Extensions

Name clarified, offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025